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Thermally-driven : Transition from hydrodynamic to Jeans escape

Alexey N. Volkov1, Robert E. Johnson1,2, Orenthal J. Tucker1, Justin T. Erwin1

1Materials and Engineering, University of Virginia, Charlottesville, VA 22904-4745

2Physics Department, NYU, NY, NY 10003-6621

Abstract

Thermally-driven atmospheric escape evolves from an organized outflow (hydrodynamic escape) to escape on a molecule by molecules basis (Jeans escape) with increasing Jeans parameter, the ratio of the gravitational to thermal of molecules in a ’s .

This transition is described here using the direct simulation Monte Carlo method for a single component spherically symmetric atmosphere. When the heating is predominantly below the lower boundary of the simulation region, R0, and well below the exobase, this transition is shown

to occur over a surprisingly narrow range of Jeans parameters evaluated at R0: λ0 ~ 2-3. The

Jeans parameter λ0 ~ 2.1 roughly corresponds to the upper limit for isentropic, supersonic

outflow and for λ0 >3 escape occurs on a molecule by molecule basis. For λ0 > ~6, it is shown that the escape rate does not deviate significantly from the familiar Jeans rate evaluated at the nominal exobase, contrary to what has been suggested. Scaling by the Jeans parameter and the

Knudsen number, escape calculations for Pluto and an early ’s atmosphere are evaluated,

and the results presented here can be applied to thermally-induced escape from a number of solar

and extrasolar planetary bodies.

1 Introduction

Our understanding of atmospheric is being enormously enhanced by extensive

spacecraft and telescopic data on the outer solar system bodies and on . The large amount of data on Titan’s atmosphere from the Cassini spacecraft led to the use of models for atmospheric escape that predicted rates that differed enormously (Johnson 2009; Johnson et al.

2009). This disagreement was due to a lack of a kinetic model for how atmospheric escape changes in character from evaporation on a molecule by molecule basis to an organized flow, referred to as hydrodynamic escape, a process of considerable interest early in the early stages of the evolution of a planet’s atmosphere (e.g., Watson et al. 1981; Hunten 1982; Tian et al. 2008) and to the evolution of (e.g., Murray-Clay et al. 2009).

The transition from hydrodynamic to Jeans escape is described in terms of the local Jeans parameter, λ = U(r)/kT(r), where U(r) is a molecule’s gravitational energy at distance r from the planet’s center, T is the local , and k is the Boltzmann constant (e.g., Chamberlain and Hunten 1987; Johnson et al. 2008). In Hunten (1982) it was suggested that if λ decreases to

~2 above the exobase, then the escape rate is not too different from the Jeans rate, but if it becomes ~2 near or below the exobase, hydrodynamic escape can occur. Since it was assumed that the transition region from Jeans to hydrodynamic escape occurred over a broad range of λ, an intermediate model, referred to as the slow hydrodynamic escape (SHE) model, was developed to describe outflow from atmospheres such as Pluto’s with exobase values λ~10

(Hunten and Watson 1982; McNutt 1989; Krasnopolsky 1999; Tian and Toon 2005; Strobel

2008a). In this model, based on that of Parker (1964) for the solar , the continuum fluid equations accounting for heat conduction are solved to an altitude where the flow velocity is still smaller than the speed of sound, and then asymptotic conditions on the temperature and density

2 are applied. Unfortunately, this altitude is often above the nominal exobase. This model,

subsequently applied to even Titan (λ~40), could overestimate the loss rate (Tucker and

Johnson 2009; Johnson 2010). Other recent models simply couple the Jeans rate to continuum

models at the exobase, even when the escape rates are quite large (Chassefiere1996; Tian et al.

2009), or they couple to a modified Jeans rate (Yelle 2004; Tian et al. 2008). Here the transition from Jeans to hydrodynamic escape from a planetary body is described via a kinetic model.

Escape from an atmosphere can be calculated from the Boltzmann equation, which can describe both continuum flow and rarefied flow (e.g., Chapman and Cowling 1970) at large distances from a planet. Here we use the Direct Simulation Monte Carlo (DSMC) method (e.g.

Bird 1994) to simulate thermally-driven flow in a one-dimensional radial atmosphere. For escape driven by heat deposited below the lower boundary of the simulation region, R0, we show that the

transition occurs over a surprisingly narrow range of Jeans parameters evaluated at R0 (λ0 ~2 –

3). That is, hypersonic outflow drives escape for λ0 ≤ ~ 2, but above λ0 ~ 3 hypersonic flow does not occur, even at large distances from the exobase, and for λ0 > ~6 the escape rate is close to the

Jeans escape rate. Following the description of the DSMC model, results for a range of λ0 are

presented with emphasis on the transition region.

2. DSMC simulations

A kinetic model for calculations of the structure of the upper atmosphere and the escape

rate is based on the Boltzmann kinetic equation. A DSMC method (Bird 1994), which is a

stochastic method for numerical solution of problems based on the Boltzmann equation, is used

here to simulate a spherically-symmetric, single component atmosphere supplied by out gassing

from a at radius R0 . This can be the actual surface with a pressure determined by the solar insolation or a virtual surface in the atmosphere, above which little additional heat is

3 deposited and at which the density and temperature are known. In such a model, escape is driven

by and heat flow from below R0 . In DSMC simulations, the real is simulated by means of large number of representative molecules of mass m. Trajectories of these molecules are calculated in a gravity field and subject to mutual collisions. It is readily shown that for a velocity independent cross section, the Boltzmann equations, and, hence, the results presented here, can be scaled by two parameters: the source values of the Jeans parameter, λ0,

1/2 -1 and a Knudsen number, Kn0 = l0 /R0 where l0 =(2 n0σ) is the of molecules at

the lower boundary with σ the collision cross section and n0 the on lower

boundary. The Knudsen number often discussed for a planet’s atmosphere is Kn(r) = l / H, where

l and H are the local mean free path of molecules and the atmospheric scale height at radial distance r. Since Kn(R0) = λ0 Kn0, it could be used, instead of Kn0, as one of the two scaling

parameters.

Simulations were conducted on a non-homogeneous mesh and the number of simulated

molecules in a was varied over a wide range (e.g., Volkov et al. 2010). Results are presented

for the hard sphere collisions, but comparisons made using variable hard spheres and forward

directed collision models (e.g.,Tucker and Johnson 2009) gave similar results. In the kinetic

model, the flow at each r is described in terms of a velocity distribution , f (r,v||,v⊥ ,t) ,

where v|| and v⊥ are velocity components parallel and perpendicular to radial direction and t is

time. Macroscopic parameters are calculated for molecular quantities Ψ(r,v|| ,v⊥ ) using the

+∞∞ integral operator Ψ = 2π Ψ(r,v ,v ) f (r,v ,v ,t)v dv dv , e.g., number density, n ∫∫ || ⊥ || ⊥ ⊥ ⊥ || −∞ 0

2 ( Ψ = 1), radial flow velocity, u (Ψ = v|| / n), parallel, T|| (Ψ = m(v|| − u) /(nk) , and

4 2 perpendicular, T⊥ ()Ψ = mv⊥ /(2nk ), with T = (T|| +2T⊥ )/3. The molecules have

gravitational energyU (r) = −GMm / r , where M is the planet’s mass and G is the gravitational

constant. The mass of the gas above R0 is assumed to be much smaller than M so that self-

gravity is neglected. The velocity distribution in the boundary cell at r=R0 is maintained to be

Maxwellian at a fixed n0 and T0 , and zero gas velocity. The exit boundary at r = R1 is placed far enough from R0 so the gas flow above this boundary can be approximately treated as

collisionless. A molecule crossing R1 with velocities v|| and v⊥ will escape if

1/ 2 2 2 v > (−2U (R1) / m) , where v = v|| + v⊥ , while a molecule with a smaller v will return to R1

with − v|| and v⊥ . Since collisions can modify the flow even at relatively large r, the effect of the position of R1 was studied (Volkov et al. 2010) and R1 is chosen to be sufficiently large in order

to eliminate the effect of upper boundary in the flow region described. The simulations were

carried out using the following fixed parameters, m, σ, T0 and R0 with λ0 and Kn0 varied by

changing M and n0. However, the results scale with λ0 and Kn0 in a sense that any two flows with

different m, σ, T0, R0, M, and n0 represent the same flow in a dimensionless form, if the λ0 and

Kn0 are the same.

Simulations are initiated by ‘evaporation’ from the cell at R0 until steady flow is reached

at large t. Steady-state flow is assumed to have occurred when the number

2 4πr n(r)u(r) = Φ varies by less that ~1% across the domain, where Φ is the escape rate. The

Jeans escape rate, ΦJeans, is defined by the upward flux of molecules with speeds

1/ 2 v ≥ (−2U (r) / m) at the nominal exobase rexo, typically determined by the scale height [l(rexo) =

H(rexo)] at large λ0 or by the curvature [l(rexo) = rexo] at small λ0. Due to the limited number of

simulation , accurately representing the tail of escaping molecules in f (r,v||,v⊥ ,t) is

5 problematic at large λ0. For objects in the solar system, λ0 varies from ~0.01 for to ~ 10

for escape from Pluto, and ~ 40 for escape from Titan, with much larger values for giant planet

atmospheres. Since the total number of simulated molecules leaving R0 in our simulations is

8 typically ~5x10 , the simulations give accurate escape rates for λ0 < 15 (Volkov et al. 2010). As

will be seen, this is more than adequate for exploring the transition region. For λ0 ≤ 10 all

simulations described below were performed with R1 / R0 = 40 , while simulations at λ0 = 15

were performed with R1 / R0 = 6 , which is enough for accurate calculation of the escape rates.

Distributions of gas parameters shown in Fig. 1 are found in simulations with R1/R0 =100.

3. Results for thermal escape

A systematic study of thermal escape was performed for values of λ0 from 0 to 15 and a

large range of Kn0 (Volkov et al. 2010), but here we describe results for small Kn0, so that the lower boundary is assumed to be well into the collision dominated region. The atmospheric properties vs. r for a number of λ0 with Kn0 = 0.001 are shown in Fig. 1. It is seen that the

dependence on r of the atmosphere density, the Mach number, and temperature are similar for

the small λ0 (0, 1, 2) cases in Fig. 1, but the of the radial dependence changes dramatically as λ0 is increased (3, 10) with a distinct transition in the narrow range between 2 and 3 as

indicated by the λ0 = 2.5 case. In particular, at a given r / R0 , n / n0 , and T⊥ /T|| tend to increase

with increasing λ0 for λ0 ≤ 2 and tend to decrease with increasing λ0 for λ0 ≥ 3. It is interesting

that T⊥ /T|| , which is considered a measure of translational non-equilibrium in expansion from a

spherical source (e.g., Cattolica et al. 1974), remains close to unity in the transition region

throughout most of the simulation domain as seen for λ0 = 2.5. The degree of translational non-

6 equilibrium, however, increases rapidly as λ0 exceeds 3. Although the perpendicular component of the temperature is found to decay to zero, as expected, at large r, it is seen in Fig. 1c that T|| does not go to zero for the larger λ0. More important to the escape problem of interest to us, the flow in Fig. 1b is seen to be hypersonic for the small values of λ0, but never becomes hypersonic for the larger λ0 shown, even at very large r well above the nominal exobase. These aspects are contrary to the assumptions that are often made in the use continuum models for escape, such as the SHE model.

100 102 (a) (b) λ =1 λ =0 0 10-1 101 0 λ0 =3 0

-2 Ma 0 n 10 10 λ0 =2 /

n λ0 =0 λ0 =2 -3 -1 10 λ0 =1 10 λ0 =10

-4 -2 λ =3 10 10 0

-3 10-5 10 Number density λ =10

0 Local Mach number Kn0 = 0.001 -4 10-6 10

-5 10-7 10 10 20 30 40 10 20 30 40 Radial distance r / R0 Radial distance r / R0

100 1.2 (c) (d) λ0 =10

λ0 =3 0 1 T / || T ||

T 0.8 / λ =2 λ =0 λ =3 ⊥ 0 0 0 T 10-1 0.6 λ0 =1

Ratio λ0 =0 λ0 =2 0.4 λ0 =1 Parallel temperature 0.2 λ0 =10 10-2 10 20 30 40 10 20 30 40 Radial distance r / R0 Radial distance r / R0

Fig. 1. Dimensionless number density n / n0 (a), local Mach number Ma = u / (5/ 3)kT / m (b), parallel temperature T|| /T0 (c), and ratio of perpendicular to parallel temperature, T⊥ /T|| , (d) vs. dimensionless

7 radial distance r / R0 at Kn0=0.001: λ 0 = 0 (red curve), 1 (green curve), 2 (blue curve), 3 (magenta curve), and 10 (brown curve). Curves for λ 0 ≤ 2 are close to that for an isentropic expansion. Simulations are fully -26 -15 scaled by λ0 and Kn0: Actual simulations were performed for N2 with fixed m=4.65x10 kg, σ=7.1x10 2 cm (collision cross-section at 90 K (Bird 1994)), T0 = 100 K, and R0 = 1000 km, while M and n0 were varied.

The atmospheric properties at very small Kn0 for λ0 ≤ 2 are found to roughly agree with

those for an isentropic expansion of a above its sonic point r* :

2 2 2 2 T T* 5 mu GMm 5 mu* GMm r nu = r* n*u* ; 2 / 3 = 2 / 3 ; kT + − = kT* + − (1) n n* 2 2 r 2 2 r*

In Eq. 1, n* , u* , and T* are evaluated at r = r* . For these small λ0, r* occurs close to R0 and hydrodynamic outflow occurs at small Kn0.A s λ0 increases from zero to ~2, the flow gradually decelerates (Fig. 1b). In this range of λ0, the thickness of the Knudsen layer decreases with

decrease in Kn0, r* ÆR0 (Fig. 1c), and T* Æ ~0.62T0 (Fig 1d), the latter is consistent with kinetic simulations of spherical expansion at zero gravity based on the Bhatnagar-Gross-Krook equation

(e.g., Sone and Sugimoto 1993). Therefore, it is seen that

2 [(5.2)kT* + mu* / 2 − GMm/ r* ]/ kT0 → 0 as λ0 Æ~2.1. This leads to rapid deceleration of the flow

as λ0 increases above 2.1. As a result, n / n0 and T|| /T0 rise up to 1 on the source surface while the local Mach number Ma rapidly drops below 0.1. The rapid transition from hydrodynamic outflow to a nearly isothermal atmosphere below the exobase is such that above λ0~2.1 the sonic point is not reached at small Kn0 up to very large distances from the source and the atmosphere is gravitationally dominated. Therefore, the transition region is not broad and λ0~10 is well beyond the region where escape by outflow can occur, unlike what has been assumed in some models for Pluto’s atmosphere (e.g., Hunten &Watson 1982; McNutt 1989; Krasnopolsky 1999;

Strobel 2008a) and even Titan (Strobel 2008b; 2009).

8

3 λ =0 λ0 =1 0 Kn0 =0.001

λ0 =2 r / R0 =10

2

λ0 =3

λ0 =10 1 Maxwellian distribution on the source surface

at temperature T0 Normalized distribution function

-2 -1 0 1 2 Parallel velocity v|| / C0

Fig. 2. Normalized distribution functions of parallel velocity v|| / C0 of gas molecules for λ 0 = 0 (red curve), 1 (green curve), 2 (blue curve), 3 (magenta curve), and 10 (cyan curve) at Kn = 0.001 and 0

r / R0 = 10 . dashed curves shows the Maxwellian distribution on the lower boundary at temperature T0 . C0 = 2kT0 / m .

In the transition region the flow velocity, u , falls close to zero at large r / R0 . Above the

transition region, the flow again accelerates at large r / R0 as only fast molecules escape. This analysis is confirmed by looking at the distributions of the parallel component of molecular

velocity, v|| , in Fig. 2. For λ0 ≤ 2 the distribution is shifted at r / R0 =10 towards large v|| and the gas velocity, u, is close to the most probable molecular velocity, while at λ0 ≥ 3 the most probable v|| is close to zero and escape is provided by the depletion of the upward moving, high- speed molecules.

9 100 Kn0 =0.0003 10-1

10-2

Φ / Φ0,0 10-3

-4

Dimensionless10 escape rate ΦJeans / Φ0,0

Transition region 10-5 0 2 4 6 8 10 12 14 Jeans parameter λ 0

FIG. 3. Escape rate Φ / Φ 0,0 (red curve with square symbols) and the Jeans escape rate Φ Jeans / Φ 0,0 evaluated at the nominal exobase (green curve with circle symbols) given as dimensionless ratio vs. Jeans 2 parameter on the source surface λ0 at Kn0 = 0.0003 . Φ 0,0 = 4πR0 n0 kT0 /(2πm) is the evaporation rate on the lower boundary. Vertical lines at λ0 = 2 and λ 0 = 2.8 indicate the transition region.

These results have implications for the variation of the molecular escape rate, Φ, with λ0.

In Fig. 3 we present a non-dimensional escape rate: the ratio of Φ to the escape rate for free

2 molecular flow from the source surface, Φ 0,0 = 4πR0 n0 kT0 /(2πm) . Remarkably, up to λ0 = 2, this ratio is very close to ~0.82, the rate found in the absence of gravity as in a -like expansion (e.g., Cong and Bird 1978; Crifo et al. 2002; Tennishev et al. 2008). The ~0.82 is due to collisions in the Knudsen layer causing the return flow. As the atmospheric outflow is choked off by the increased gravitational binding, Φ/Φ0,0 rapidly drops a couple of orders of magnitude between λ0 = 2 to 3. In this transition region, escape is a product of a rapidly decreasing fraction

10 of the velocity distribution function with v >vesc (e.g, Fig.2), assisted by a rapidly decreasing flow speed (e.g. Fig.1b). Therefore, for λ0 ≥ 6, it is seen that the escape rate is close to the Jeans rate. From Fig. 3 the ratio of the escape rate to the Jeans estimate varies from 1.7 to 1.4 as λ0 goes from 6 to 15, consistent with earlier results (Tucker and Johnson 2009). A modified Jeans escape rate, accounting for the non-zero outflow gas velocity, u, can provide a good approximation for the actual escape rate at λ0 ≤ 6 as it is described in Volkov et al. (2010).

In scaling the escape rates using λ0 and Kn0, care must be taken. Unlike for the case λ0≤2,

where Φ / Φ 0,0 →~ 0.82 at Kn0 Æ0, it is seen in Fig. 4 that escape rates at λ0 > 3 for

Kn0 = 0.0003 cannot necessarily be consider a good approximation to the escape rate in the

limit Kn0 → 0 .

101 λ =0

0,0 0 100 Φ

/ λ0 =1

Φ -1 10 λ0 =3

10-2

10-3

λ0 =10 10-4

-5 10 λ0 =15 Dimensionless escape rate

10-6 10-4 10-3 10-2 10-1 100 101 Knudsen number Kn 0

Fig. 4. Dimensionless escape rate Φ / Φ 0,0 v.s. Knudsen number Kn0 at λ 0 = 0 (black curve), 1 (red 2 curve), 3 (green curve), 10 (blue curve), and 15 (magenta curve). Φ 0,0 = 4πR0 n0 kT0 /(2πm) is the evaporation rate on the lower boundary.

11 In Fig. 5 DSMC results are also given for the thermal flux through the atmosphere for λ0

= 10, a value relevant to Pluto’s present atmosphere and considered an intermediate case between a comet outflow and a terrestrial-like atmosphere. It has been argued that, at such values of λ0, escape driven by thermal conduction can be continued into the exobase region and above until the flow speed is some fraction of the speed of sound (e.g. Strobel 2008b). This picture has been shown to be incorrect (Johnson 2010). Here we note also that the flow speed does not reach the speed of sound at small Kn0 (Fig. 1c), and from Fig. 5 it is seen that even a couple of scale heights below the exobase the heat flux is not well described by the Fourier law,

−κ(T)dT / dr , where the thermal conductivity κ(T) is defined by the first approximation of the

Chapman-Enskog method (Chapman & Cowling 1970) and T is the actual temperature found in the DSMC simulations. This law drastically overestimates the energy flux above an exobase defined by neutral-neutral collisions, consistent with λ0 = 10 being well above the critical Jeans parameter.

12 10-2

λ0 =10 0 q

/ -3

q 10

flux Fourier q / q0 , t Kn0 = 0.001 10-4

Actual q / q0 , Exobase free molecular flow 10-5 Actual q / q0 , Dimensionless hea Kn0 =0.001

10-6 1 2 3 4 5 6 7 8 9 Radial distance r / R0

Fig. 5. Dimensionless heat flux q / q0 vs. radial distance r / R0 calculated for λ 0 = 10 at Kn0 → ∞

(free molecular flow: black dotted curve) and Kn0 = 0.001 (red dashed curve) . Solid curve represents dimensionless Fourier heat flux, where q = −κ(T )dT / dr is calculated based on temperature T (r) found in DSMC simulations at Kn0 = 0.001: Thermal conductivity κ(T ) is given by the first approximation of the Chapman-Enskog method for a hard sphere gas (Chapman & Cowling 1970) and

q0 = n0kT0 2kT0 / m .

The results in Fig. 3 and 4 can now be used to evaluate previous calculations of atmospheric mass loss rates. For Pluto’s predominantly atmosphere, when all of the heating is assumed to be below R0 = 1450 km, the Q=0 case in Strobel (2008a), then λ0 ~ 23.

From Fig. 3 the escape rate is seen to be very close to the Jeans rate and much smaller than 10-5 times the surface flux, Φ0,0. Therefore, the actual the mass loss rate is orders of magnitude below the rate estimated in that paper (~5x1028 amu/s). Of more interest, is the mass loss rate from

Pluto at solar medium heating: in Strobel (2008a) Kn0 = 0.01 occurs at an R0~3600 km, which is

7 -3 well above the solar heating maximum and corresponds to n0 ~3.8x10 cm and T0 ~ 83K

13 27 resulting in λ0 ~ 10. Based on the results in Fig.4 the mass loss rate would be ~2x10 amu/s.

This is also more than an order of magnitude below that predicted: ~9x1028 amu/s. Therefore, it is clear that modeling in support of the New Horizon mission to Pluto will require iteratively coupling a kinetic model of escape to a fluid description of the lower atmosphere (Tucker et al

2010). As a second example, in a study of the EUV heating of the Earth’s atmosphere, Tian et al.

(2008) found the onset of hydrodynamic escape of , and the resulting adiabatic cooling of the thermosphere, occurs at an exobase having a value of λ ~ 5.3. Since this altitude corresponds to a Kn0 ~ 0.2, and is well above the heating peak, the inferred onset is in disagreement with the results presented here. Since, they used a modified Jeans rate, their overestimate of the escape rate is only about a factor of 4-5. What may be more important is the effect on their description of the atmospheric structure near the exobase. Therefore, it is clear that fluid calculations of atmospheric escape should be tested against the results presents here, which by scaling can be used as upper boundary conditions.

Summary

A kinetic model for planetary atmospheres was used here to study the transition from hydrodynamic escape to escape on a molecule by molecule basis. When heat is deposited primarily below the lower boundary of the simulation region, R0, and the collisions can be described by a hard sphere model, then the results presented can be fully scaled by two parameters evaluated at R0: the Jeans parameter, λ0, and the Knudsen number, Kn0 (or Kn(R0)). In the collision dominated regime at small Kn0, this transition is found to occur over a surprisingly narrow range of λ0, unlike what has been assumed in some continuum models of thermal escape.

That is, below a critical value of λ0 (~2.1), hydrodynamic outflow occurs, and is roughly described by an isentropic expansion starting from the sonic surface. Above the transition

14 regime, λ0 > ~3, escape occurs on a molecule by molecule basis and for λ0 > ~6 we show that the escape rates do not deviate enormously from the Jeans rate. Although the simulations described here are highly idealized (hard sphere cross section and single component, spherically symmetric atmosphere) the overall conclusions will apply to the simulation of more complex atmospheres which are in progress. Therefore, re-evaluation of a number of studies, such as those for the early terrestrial atmospheres and the atmosphere of Pluto, soon to be visited by the New Horizon spacecraft, must be carried out using kinetic models of the upper atmosphere.

Acknowledgements

This work was supported by grants from NASA’s Planetary Atmospheres Program and the

Cassini Data Analysis Program.

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